We can make use of the fact that concentrations lend themselves to
encoding probabilities (an idea also used in
[1]}) to derive a non-sequential version of
the same computation. Consider two proteins, an Activator and a
Repressor, that have expression levels \(c_1\)  and \(c_2\) , respectively
(Fig. REF B). We can normalize and sample from the
(unnormalized) vector \(\theta = [c_1\ c_2]\)  by having the
Activator and Repressor bind, in mutually exclusive fashion, a third molecule which we denote
Integrator. If Integrator is not in excess of
Activator and Repressor, then the fraction of Activator-bound Integrator will be proportional to the
concentration of Activator normalized by the sum of
concentrations of Activator and Repressor: \(\frac{[Act]}{[Act] +[Rep]}\)  (Fig. REF B). This means that a molecular
interaction that depends on the concentration of Activator-bound
Integrator will occur in proportion to the probability
\(\frac{c_1}{c_1 + c_2}\) . Note that this does not rely on sequential order; the
interactions between Activator, Repressor, and Integrator can all take place
in parallel. Normalization of protein abundances in this manner is conceptually
similar to the “divisive normalization” operation that has been
observed in various neural circuits [2]}.
